Optimal. Leaf size=44 \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4627, 4681, 29} \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 4627
Rule 4681
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x^3} \, dx &=-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a \int \frac {\sin ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.00 \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 44, normalized size = 1.00 \[ \frac {2 \, a^{2} x^{2} \log \relax (x) - 2 \, \sqrt {-a^{2} x^{2} + 1} a x \arcsin \left (a x\right ) - \arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 82, normalized size = 1.86 \[ \frac {1}{2} \, {\left ({\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \arcsin \left (a x\right ) + 2 \, a \log \left ({\left | x \right |}\right )\right )} a - \frac {\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.98 \[ -\frac {\arcsin \left (a x \right )^{2}}{2 x^{2}}-\frac {a \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x}+a^{2} \ln \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 40, normalized size = 0.91 \[ a^{2} \log \relax (x) - \frac {\sqrt {-a^{2} x^{2} + 1} a \arcsin \left (a x\right )}{x} - \frac {\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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